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1、2014b9?pJp 33y. 3?cosx 1 x2 2?x =A 2?1 + cosA 2 A2 AA 8?:LHS 2 A+2 B+2 C 818 8 333KKK1.8. 3b?n?/ABC,y3 4sinA+sinB+sinC+3 2tanA+tanB+tanC 21+ 2y. d?:LHS 2q22(sin A+sin B+sin C) 3+tan A+tan B+tan C3|?2sinx + tanx 3x, 2 x 0?:2(sinA + sinB + sinC) + (tanA + tanB + tanC) 3?yps:?a,b,c 0,x (0, 2)y2r a2+ b2
2、+ c2 3sinx x+a + b + c 3tanx x a + b + cKKK1.9. 3n?/ABC,e2A + 3B = y4(a + b) 0KKK1.10. 3n?/ABC,?u45o,y:cotA + cotB + cotC + 3cotAcotB cotC 6 4(2 2)y. ”?A = min(A,B,C)Kk46 A 6 3|cotx + coty =1 cotxcoty cotB + cotC?cotB cotC = 1 cotA(cotB + cotC)?:LHS = cotA+cotB+cotC+3cotA(1cotA(cotB+cotC) = 4cotA+(c
3、otB+cotC)(13cot2A)d?u45oTn?/7,b?n?/5?:(cotx)00= 2cotxcsc2x 0,x (0, 2)Kdjensen?k:cotB + cotC 2cotB + C 2 (1 3cot2A 6 0eIy4cotA + 2cotB + C 2(1 3cot2A) 6 4(2 2)4ePtanA 2= tKdA?t 2 1,13 K?LHS =4 3(1 t2)2 2t6 4(2 2)KKK1.11. 3n?/ABC,y(sinA)sinB+ (sinB)sinC+ (sinC)sinA 1.19y. d|?:(sinA)sinB=1 (1 +1 sinA 1
4、)sinB1 1 + (1 sinA 1)(sinB)sinA sinA + sinB-tanA 2= a,tanB 2= b,tanC 2= cKab + bc + ca = 1 K:LHS XsinA sinA + sinB=Xa(1 + b2) a(1 + b2) + b(1 + a2)2|1 + b2= ab + bc + ca + b2= (b + a)(b + c)?:Xa(1 + b2) a(1 + b2) + b(1 + a2)=Xa(a + b)(b + c) a(a + b)(b + c) + b(b + a)(a + c)=Xab + ac1 + ab2d?Xab + a
5、c1 + ab=Xab 1 + ab+Xac 1 + ab1 1 +Pa2b2+1 1 +Pa2bc=1 1 +Pa2b2+2 3 Pa2b2(1 +2)24 RHSKKK1.12. 3n?/ABC,ef?(n?)sinA 2nrsinB 2sinC 2y. dz?,?= sinA 2ns cosBC 2 cosB+C 2 26 sinA 2nr 1 2(1 sinA 2)-sinA2= aKd?nr1 2nan(n na) 6n(n + 1)np2(n + 1)5KKK1.13. O?n?/?/G(dn?/n?1. cosA = sinB + sinC 3 22. sin(B A)sinC
6、+ sinA + cosB =3 23. cos2A +3(cos2B + cos2C) +52= 04.sinA 1=sinB3=sinC 25. cos2A + 22cosB + 22cosC = 36. sinA + sinB + sinC =3 +327. sinA 2cosB 2cosC 2+ sinB 2cosC 2cosA 2+ sinC 2cosA 2cosB 2=9 8y. 5?sinA 2cosB 2cosC 2=1 2sinA 2(cosB + C 2+ cosB C 2)=1 2sin2A 2+1 2cosB + C 2cosB C 2=1 cosA 4+cosB +
7、cosC 4?:cosA + cosB + cosC =3 2?A = B = CKKK1.14. tanA 2,tanB 2vx2a1x+b1= 0;tanB 2,tanC 2vx2a2x+b2=0 ;tanC 2,tanA 2vx2 a3x + b3= 0,y(1 a1+ b1)(1 a2+ b2)(1 a3+ b3) 6 5616 32403y. 5?:(x2+ a1x + b1)(x2+ a2x + b2)(x2+ a3x + b3) = (x tanA 2)2(x tanB 2)2(x tanC 2)2-x = 1,eIy:(1 + tanA 2)2(1 + tanB 2)2(1 +
8、 tanC 2)26 5616 324035?:tanB + C 4= tan A 4X tanA 4+X tanA 4tanB 4= 1 + tanA 4tanB 4tanC 4K:(1 + tanA 2)2(1 + tanB 2)2(1 + tanC 2)2= 4(1 + tanA 4tanB 4tanC 4)26duA2.B2,C 2 (0, 4)?tanA 4tanB 4tanC 4 (0,1)PtanA 4tanB 4tanC 4= x3?:1 + x3= 1 + tanA 4tanB 4tanC 4=X tanA 4+X tanA 4tanB 4 3x + 3x2 (x + 1)(
9、x 2 +3)(x 2 3) 0 0 2,:I =Xk=01 F2ky. ?Fn=15(5 + 12)n (5 12)n)Pa =5 + 12K1 F2k=5a2n a2n2-a2n= x?: 1 F2k=5xx2 1=5(1 x 11 x2 1)?:I =Xk=01 F2k= I =5Xk=0(1 a2n1 a2n+1 1) =5a 1=7 52KKK1.16. a,b,c,d 0,vsina + 7sinb = 4(sinc + 2sind);cosa + 7cosb = 4(cosc + 2cosd)y:2cos(a d) = 7cos(b c)y.KKK1.17. y(4cos29o
10、3)(4cos227o 3) = tan9oy. |cos3x = 4cos3x 3cosx?cos3xcosx= 4cos2x 3-x = 9,x = 27?y7KKK1.18. 3n?/ABC,sinA + sinB + sinC 6 1,y:minA + B,B + C,C + A B CKeIyB + C X sinA 2sinA sinA 60K7,A 150 B + C 1y. n8B,un = 15?:|cosa1| + |sina1| cos2a1+ sin2a1= 11en?y,eyn + 1?: d8Bb?,Iy|sinan+1| + |cos(a1+ . + an+1)|
11、 |cos(a1+ a2+ . + an)|-skak?,5?|cossn| = |cos(sn+1 an+1)| = |cossn+1cosan+1+ sinsn+1sinan+1|6 |cossn+1cosan+1| + |sinsn+1sinan+1|le|cossn+1| + |sinan+1|?8BKKK1.23. lm2 62,2 +62 ?o,y7,3a,bv|ap 4 b2 bp 4 a2| 6 2y. -a = 2sinx,b = 2sinyK|ap 4 b2 bp 4 a2| = 4|sin(x y)| |sin(x y)| 6 sin 65?m2sin15,2sin75K
12、x,y 15,75?nm15,15,15,45,45,75?o7,k3mS,d?u30KKK1.24. n?/ABCS?u120o,ycosA + cosB cosC sinA + sinB sinC 33y. ?En?/ABC,PA0= 120oA,B0= 120oB,C0= 120oCKdA0B0C0n?/,Kkn?sinA0+ sinB0 sinC032(cosA + cosB cosC) +1 2(sinA + sinB sinC) 05?sinA + sinB sinC 0?n?KKK1.25. ?anva1= t,an+1= 4an(1 an),n 1k?t?a2004=0?9y.
13、 4?dan+1 1 = (2an 1)2“2an 1 = cosbnKsin2bn= sin2bn+1 2 bn+1= 2bndu4x(1 x) (0,1)?0 6 x 6 1oa2004= 0Kt (0,1)K”?a1= sin2ao?a2004= sin222003a a =k 22003 t = sin2k 22003 ?k = 0,1,2.,22002?22002tKKK1.26. a0=2 +3 +6,a n+1=a2n 5 2(an+ 2),n 0yan= cot2n3 3 2y. bn= an+ 2?:bn+1=b2n 1 2bn2“bn= cotcn?cotcn+1= cot
14、2cn25?:cot 24=cos 24 sin 24=2cos2 24 2sin 24cos 24=1 + cos 12 sin 12=1 +24+646424= 2 +2 +3 +6 = a 0+ 2?c0= 24Kcn=2n3 3?KKK1.27. a,b,c 0, 2,ysinasin(a b)sin(a c) sin(b + c)+sinbsin(b c)sin(b a) sin(c + a)+sincsin(c a)sin(c b) sin(a + b) 0y. |n?sin(x + y)sin(x y) =cos2y cos2x 2=1 2sin2y 1 + 2sin2x 2= sin2x sin2yK?dXcycsina(sin2a sin2b)(sin2a sin2c) 0-sina = x,sinb = y,sinc = z?dyXcycx(x2 y2)(x2
